A Survey of Test Ideals

doi:10.1515/9783110278606.39

Cited by 24 publications

(22 citation statements)

References 112 publications

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“…In this section, we provide the basic definitions and results from the theory of test ideals and Cartier algebras. A more complete account of these theories may be found in the surveys, [ST11,BS13].…”

Section: Background On Cartier Algebras and Test Idealsmentioning

confidence: 99%

A new subadditivity formula for test ideals

Smolkin

2020

Journal of Pure and Applied Algebra

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We exhibit a new subadditivity formula for test ideals on singular varieties using an argument similar to [DEL00] and [HY03]. Any subadditivity formula for singular varieties must have a correction term that measures the singularities of that variety. Whereas earlier subadditivity formulas accomplished this by multiplying by the Jacobian ideal, our approach is to use the formalism of Cartier algebras [Bli13]. We also show that our subadditivity containment is sharper than ones shown previously in [Tak06] and [Eis10]. The first of these results follows from a Noether normalization technique due to Hochster and Huneke. The second of these results is obtained using ideas from [Tak08] and [Eis10] to show that the adjoint ideal J X pA, Zq reduces mod p to Takagi's adjoint test ideal, even when the ambient space is singular, provided that A is regular at the generic point of X. One difficulty of using this new subadditivity formula in practice is the computational complexity of computing its correction term. Thus, we discuss a combinatorial construction of the relevant Cartier algebra in the toric setting.

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Section: Background On Cartier Algebras and Test Idealsmentioning

confidence: 99%

A new subadditivity formula for test ideals

Smolkin

2020

Journal of Pure and Applied Algebra

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“…3]. We refer to [ST12] for a more comprehensive overview of the theory of test ideals. Let X be a smooth variety over a perfect field k of characteristic p > 0.…”

Section: Asymptotic Test Idealsmentioning

confidence: 99%

Continuity of Plurisubharmonic Envelopes in Non-Archimedean Geometry and Test Ideals

Gubler¹,

Jell²,

Künnemann³

et al. 2019

Annales De l'Institut Fourier

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Let L be an ample line bundle on a smooth projective variety X over a non-archimedean field K. For a continuous metric on L an , we show in the following two cases that the semipositive envelope is a continuous semipositive metric on L an and that the non-archimedean Monge-Ampère equation has a solution. First, we prove it for curves using results of Thuillier. Second, we show it under the assumption that X is a surface defined geometrically over the function field of a curve over a perfect field k of positive characteristic. The second case holds in higher dimensions if we assume resolution of singularities over k. The proof follows a strategy from Boucksom, Favre and Jonsson, replacing multiplier ideals by test ideals. Finally, the appendix by Burgos and Sombra provides an example of a semipositive metric whose retraction is not semipositive. The example is based on the construction of a toric variety which has two SNC-models which induce the same skeleton but different retraction maps.

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“…We now define test ideals for F -finite schemes of prime characteristic p > 0. See [ST12] and [TW18, §5] for overviews of the theory. We take Schwede's characterization of test ideals via F -compatibility [Sch10] as our definition.…”

Section: Base Locimentioning

confidence: 99%

The Gamma Construction and Asymptotic Invariants of Line Bundles Over Arbitrary fields

Murayama

2019

Nagoya Math. J.

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We extend results on asymptotic invariants of line bundles on complex projective varieties to projective varieties over arbitrary fields. To do so over imperfect fields, we prove a scheme-theoretic version of the gamma construction of Hochster and Huneke to reduce to the setting where the ground field is F -finite. Our main result uses the gamma construction to extend the ampleness criterion of de Fernex, Küronya, and Lazarsfeld using asymptotic cohomological functions to projective varieties over arbitrary fields, which was previously known only for complex projective varieties. We also extend Nakayama's description of the restricted base locus to klt or strongly F -regular varieties over arbitrary fields. X ; see [PST17]. For imperfect fields, however, this approach runs into another problem:(III) Most applications of Frobenius techniques require k to be F -finite, i.e., satisfy [k : k p ] < ∞.This last issue arises since Grothendieck duality cannot be applied to the Frobenius if it is not finite. Recent advances in the minimal model program over imperfect fields due to Tanaka [Tan18; Tan] suggest that it would be worthwhile to develop a systematic way to deal with (III).Our first goal is to provide such a systematic way to reduce to the case when k is F -finite. While passing to a perfect closure of k fixes the F -finiteness issue, this operation can change the singularities of X drastically. To preserve singularities, we prove the following scheme-theoretic version of the gamma construction of Hochster and Huneke [HH94].Theorem A. Let X be a scheme essentially of finite type over a field k of characteristic p > 0, and let Q be a set of properties in the following list: local complete intersection, Gorenstein, Cohen-Macaulay, (S n ), regular, (R n ), normal, weakly normal, reduced, strongly F -regular, F -pure, Frational, F -injective. Then, there exists a purely inseparable field extension k ⊆ k Γ such that k Γ is

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A Survey of Test Ideals

Cited by 24 publications

References 112 publications

A new subadditivity formula for test ideals

A new subadditivity formula for test ideals

Continuity of Plurisubharmonic Envelopes in Non-Archimedean Geometry and Test Ideals

The Gamma Construction and Asymptotic Invariants of Line Bundles Over Arbitrary fields

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